ME 105 – Mechanical Engineering Laboratory
Spring Quarter 2010
Experiment # 3: Pipe Flow
Objectives: a) Calibrate a pressure transducer and two different flowmeters (paddlewheel and orifice plate); b) Use the flowmeter and pressure transducer to measure the friction factor for pipes of different diameter, of different lengths, and for different flow rates. Check for Reynolds number scaling and compare with the Moody diagram; c) Measure minor losses in fittings and compare with empirical rules of thumb; d) Use a hydraulic analog of a Wheatstone bridge to test rules of thumb for minor losses.
Introduction
Volumetric flow rate, pressure, and head losses are key fundamental quantities in analyzing and designing piping systems. This experiment will introduce you to basic measurement techniques and to some principles of pipe flow. In this experiment three basic devices – a pressure transducer, an orifice‐plate flowmeter and a paddlewheel flowmeter ‐ are calibrated and compared against standard practice, and then used to make fundamental measurements of losses in pipes, fittings, and piping networks.
Pre-Lab Reading
Review relevant material from your undergraduate fluid mechanics courses, including (i) Reynolds number, (ii) losses in straight pipes and the Moody diagram, (iii) Bernoulli’s equation and the mechanical energy balance, (iv) orifice meters, and (v) minor losses in fittings. Some of this material is presented below, but this lab handout is not a substitute for more extensive
background reading.
Pre-Lab Work
Prepare and submit an outline that includes: Calibrations to perform
Data sets to collect
Possible sources of experimental uncertainty and a plan for quantifying these errors Brief description of the work plan
Any equations or physical parameters that may be needed during the laboratory session
(See general lab guidelines & print out grading sheet from website).
PreLab Exercises:
1. Hydrodynamic losses in pipe flow are characterized by measuring the pressure drop P over a length of pipe L. If you anticipate using flow rates of 0.5 gals/min through 1/4” i.d. smooth-wall tubing, and want a pressure drop of 10 kPa, what length, L, of tubing should you use? Express your answer in meters. Note: You will note that this problem statement uses mixed units, which unfortunately are a fact of life in engineering calculations. You should know how to do unit conversions accurately and quickly. A good rule of thumb is to convert all units to SI before doing any numerical calculations.
Hint: Assume that the working fluid is water at 20°C, and refer to a standard Moody
diagram to complete this task.
2. The Validyne pressure transducer measures pressure differences between the two sides of a stainless steel plate (diaphragm). It will be calibrated by applying hydrostatic pressure to one side. If water at 20°C is the working fluid, what range of water heights should be used to calibrate the device over a range of differential pressures from 0‐20 kPa?
3. The kit includes 1/8”, 1/4”, and 3/8” i.d. tubes. If water at 20°C is the working fluid and the transition Reynolds number is taken as 2,000, calculate the velocity and the volumetric flow rate for transition from laminar to turbulent flow for each sized tube. Record both velocities and volumetric flow rates in your notebook for future reference.
4. The paddlewheel flowmeter works on the principle that the oncoming flow rotates the paddlewheel at a frequency that is related to the flow rate. There will be some backflow as the vane of the paddlewheel sweeps forward. Consider the hypothetical situation where the flow rate vs. frequency relation is exactly linear. What would that tell you about the backflow?
5. In a standard fluids text we find the following “rules of thumb” for the ratio of equivalent length to pipe diameter Le/D for minor losses due to:
Le/D
Standard elbow: 30
Standard tee: flow through run 20 flow through branch 60
Consider the flow of water at Q = 2 liters/min. through a ¼” diameter tube containing an elbow. Use the “rule of thumb” to estimate the pressure drop across the elbow. Express your answer in Pascals.
6. Referring to the pipe network shown in Figure 1, and with the aid of the development in the handout, manipulate the energy balance to obtain a working equation for the head losses as follows.
A) If p23 p2p3 , i.e. the bridge is ‘balanced’, and (as is true of our setup), all the 0 tubing between point 1 and points 2 and 3 is the same diameter and length, the fittings are identical, and the elevation at points 2 and 3 are the same, what is the left hand side of equation (13)?
B) Now if in addition the diameter of the tubing at the outlets 4 and 5 is identical what is the relationship between u4 and u ? 5
C) With all this in mind, if the outlets 4 and 5 are held such that the water exits into the atmosphere, what is the working equation relating the elevations at 4 and 5 and the losses in legs A and B?
Figure 1: A simple pipe network equivalent to a Wheatstone bridge.
Equipment
• Omega paddlewheel flowmeter • Validyne pressure transducer with “bleeding” screwdriver • Water reservoir and sump‐pump • Teflon tubing and fitting assortment • Flow needle valve • Orifice plate • Bucket • Balance • Stopwatch Oscilloscope and power supply Thermometer Pump 1 2 3 LA LB
Technical Data
Orifice plateUpstream pipe diameter = 9.53 mm Orifice diameter = 4.76 mm
System Description
You will need to set up a simple method to calibrate a pressure transducer by providing a known pressure difference between the two sides of the transducer. In addition you will have to construct a “water-bench” to perform measurements that allow you to calibrate the pressure transducer and the two kinds of flowmeters, and investigate head loss in pipe flow, fittings, and pipe networks. Although you will decide the specific arrangement, Fig. 2 shows generically the layout of the flow loop.
Figure 2: Flow-loop schematic.
Theoretical orifice relations
An orifice plate is one of the most common flow measurement devices. Using a control volume approach shown in Fig. 3, it is possible to obtain an expression for the flow coefficient in terms of the flow rate Q, the pressure difference P1‐P2 across the orifice plate, and the geometrical parameters of the flowmeter. Applying conservation of mass for steady flow,
pump
needle
valve
paddlewhee
flow meter
return
water supply
orifice plate
pipe section
pressure
transducer
A1V1 A2V2, (1)
and Bernoulli’s equation between position 1 to position 2,
2 2 2 2 1 1 2 1 2 2 gZ P V gZ P V , (2)
Figure 3: Flow in the vicinity of an orifice plate.
we find that if Z1=Z2:
2
1 2 2 2 2 1 1 ( / ) 2 A A V P P , (3)where V is the flow velocity, A is the area, is the density, g is the acceleration due to gravity and Z is the elevation. This can be rewritten in terms of the volumetric flow rate as a function of the pressure difference:
2
1 2 2 1 2 2 2 ) / ( 1 ) ( 2 A A P P A A V Q . (4) For the orifice‐plate meter shown in Fig. 3, the area A2 is not given by the orifice diameter d,but rather the diameter of the vena contracta, (where the flow has a minimum cross‐ sectional area). This area is unknown and will change with the flow rate. Consequently, (4)
D
d
Orifice plate Pipe
Control volume 1 2 Differential pressure transducer
is often written withA2 d2 /4, and a “discharge‐coefficient” CD is added to account for the combination of these geometric effects and viscous losses: 1 2 2 4 2 ( ) 1 D P P Q C A , (5) where d /D, where d is the orifice diameter and D is the diameter of the pipe. The
discharge coefficient for an orifice‐plate meter is not constant and is found experimentally by measuring both Q and (P1P2) and applying equation (5). Values for CD have been
measured for standardized tap locations, which allow flow rates to be measured from a pressure drop across the orifice plate. Figure 4 shows the typical dependence of CD as a function of geometry and Reynolds number, Re. Figure 4: Discharge coefficient curves for a standard orificeplate flowmeter.
Head Loss in Pipe Flows
There is a pressure drop when a fluid flows in a pipe because energy is required to overcome the viscous or frictional forces exerted by the walls of the pipe on the moving fluid. In addition to the energy lost due to frictional forces, the flow also loses energy (or pressure) as it goes through fittings, such as valves, elbows, contractions and expansions. This loss in pressure is often due to the fact that flow separates locally as it moves through such fittings. The pressure loss in pipe flows is commonly referred to as head loss. The frictional losses are referred to as major losses
(hl) while losses through fittings, etc, are called minor losses (hlm). Together they make up the
Mechanical Energy Equation for Pipe Flows
The mechanical energy equation between any two points 1 and 2 for steady incompressible flow is: lT h gz V P gz V P 2 2 2 2 1 2 1 1 2 2 . (6)
(It also be noted that for flow without losses, hlT = 0, and the energy equation reduces to Bernoulli’s Equation.) The terms in parentheses represent the mechanical energy per unit mass at a particular cross-section in the pipe. Hence, the difference between the mechanical energy at two locations, i.e. the total head loss, results from the conversion of mechanical energy to thermal energy due to frictional effects.
For an incompressible flow, conservation of mass determines V2 (since, V1A1 V2A2) and so the
terms involving the fluid velocity are determined by geometry. If the elevation at position 2 is known, the change in the gravitational potential is known. The net result is that if the pipe diameter is constant and the elevation does not change, the head loss is manifested simply as a pressure loss.
Major Losses
The major head loss in pipe flows is expressed in the following way: 2 2 V D L f hl , (7)
where L and D are the length and diameter of the pipe, respectively, and V is the average fluid velocity through the pipe. This may be taken as a definition of the friction factor, f. In general, the friction factor is a function of the Reynolds number Re and the non-dimensional surface roughness /D, and is determined experimentally. The plot of f vs. Re is usually referred to as the Moody Diagram, after L. F. Moody who first published this data in this form.
Minor Losses
The head losses associated with fittings such as elbows, tees, couplings, etc. are referred to as “minor losses”. In some cases, such as short pipes with multiple fittings, these losses are actually a large percentage of the total head loss and hence are not really “minor”. Minor losses are expressed as either 2 2 V K hlm , (8a)
where K is the Loss Coefficient and must be determined experimentally for each situation, or as
2 2 e lm L V h f D , (8b)
wherein the loss is expressed in terms of the (known) friction factor and an equivalent Le/D . For example, an elbow creates a loss that is roughly equivalent to a pipe of length of 30 pipe diameters (see the table in Prelab Question 4). Loss coefficients, K and/or equivalent length
ratios /Le D can be found in a variety of handbooks: data for specific simple fittings are available in most undergraduate Fluid Mechanics texts.
Pipe networks: the hydraulic analog of a Wheatstone bridge
Consider the pipe system shown schematically in Figure 1. We are interested in describing the pressure loss through all the legs of this simple network. If both legs A and B exit into the atmosphere, then the pressure differentials downstream of junctions 2 and 3 can be defined as:(9)
where pa is atmospheric pressure. The energy equation for these two branches yields:
2 2
and (10a,b)
2 2
Assume that leg A of the network consists of only a straight tube uniform tube and therefore the head loss h is LA 2 2 A A LA A A L V h f D . (11)
The head loss in leg B of the network, h , includes losses through the pipe itself but also any LB minor losses due to the insertion of elbows, etc. In general,
2 2 2 2 eB B B B LB B B B B L L V V h f f D D , (12)
where we have chosen to express the minor losses in terms of equivalent pipe lengths, LeB. Subtracting (10a) from (10b) we obtain:
2 2 13 2 3 and A a B a p p p p p p
This particular pipe network is analogous to a Wheatstone bridge. The purpose of the electrical version of such a bridge is to be able to measure small changes in resistance accurately. In the hydraulic analog we measure small changes in head loss. When the bridge is balanced, i.e. there is no flow through the leg L23, the pressures at points 2 and 3 must be the same: (otherwise, the
pressure gradient would drive a flow through the leg).
In our laboratory setup, the leg L23 viewed from the side is shaped in an arc as shown in Figure 5.
A small tightly fitting sphere is placed in the tube. Any flow in the leg will exert a drag on the sphere and it will rise above the center. By contrast, a no flow condition will result in the sphere positioned at zero degrees from the vertical sincep23 p2 p3 . Thus, monitoring the sphere 0 position allows a coarse measurement of bridge balance.
Figure 5: Schematic of the sphere in the arched tube comprising the center leg.
Experimental Procedure:
General
You will be making a variety of measurements with water, the physical properties of which are temperature dependent. For this reason, it is very important that you know the temperature of
the water for each measurement.
Week One
1) Calibration of the pressure transducer
The first step is to calibrate the output voltage from the Validyne differential pressure transducer. Apply known pressure differences to the two sides of the transducer using hydrostatic pressure. Five to ten data points should be obtained, ranging from a zero pressure differential to a pressure differential of about 20 kPa. Perform a linear least‐ squares analysis of the data before week two. Try both linear and a quadratic fits and compute the goodness of fit. If a linear fit is sufficiently accurate, record the slope and intercept of the resulting line for later use in data acquisition.
2) Calibration of the paddlewheel and orifice plate flowmeters
Paddlewheel flowmeter: The paddlewheel flowmeter outputs a pulse train whose frequency is related to the flow rate. Calibrate the paddlewheel using varying flow rates by measuring the frequency as a function of flow rate. Five to ten data points should be obtained. Determine the rising and falling cutoff flow rates, i.e. the discharge below which the paddlewheel is motionless or erratic. You will notice that at low flow rates the frequency is erratic, and that the frequency fluctuates at all flow rates. Do your best to get an average reading from the oscilloscope. What might cause such fluctuations? Orifice plate flowmeter: Measure the pressure drop across the orifice plate as a function of flow rate. Five to ten points should be obtained. These data will be used to determine the discharge coefficient CD for the orifice plate as a function of the Reynolds number.
3) Investigation of major losses
Prepare 6’ – 8’ lengths of the three different diameter tubing. Using the paddlewheel to measure flow rate and “Tees” as pressure taps, obtain data for pressure drop over a given length as a function of flow rate for the three different sized tubes. Since the Moody diagram is for long, straight tubes, try to make your tube runs as long and as straight as feasible. Obtain 5-10 data points for each tube over the maximum range of flow rates possible. These data will be used to determine the friction factor, f, as a function of the Reynolds number. These values will also be compared to the standard Moody diagram, so you should perform calculations on some of the data during the experiment to make sure the comparison is reasonable. Complete these calculations and the comparison with the Moody diagram before week two.
Week Two
4) Major losses
Depending on the quality of your data from week one, you may chose to check calibrations and/or repeat your measurements of major losses.
5) Investigation of minor losses
Using the paddlewheel to measure flow rate and “tees” as pressure taps, measure the minor losses for an elbow, a tee, and a straight coupling as a function of flow rate. Use only one diameter tube and make sure your data are taken in the turbulent regime.
6) Investigation of a simple pipe network
Set up the flow system shown in Figure 1 of this handout incorporating the section of arced tubing between the points 2 and 3. Use ¼” tubing for these experiments. It is best to place a needle valve before the branch so that the flow rate can be controlled.
a) Prepare two 6’ long lengths of ¼” tubing to serve as legs A and B. Set the flow rate with the needle valve in the midrange of the pump and make sure that the tube exits are at the same elevation. Curiously, although the two legs are identical tubing and identical lengths, the bridge may be slightly out of balance. This could be due to a number of factors, including different coiling of the two legs, burrs and rough edges where the tube was cut, slight differences in the
losses in the two elbows at points 2 and 3, etc. By raising or lowering the exit tubes, determine which of the branches has the larger loss, and shorten the appropriate tube in order to bring the bridge into balance.
b) Experiment with the effect of raising or lowering one tube exit elevation on the bridge balance. In this way, you will obtain some feeling for the ‘response time’ of the middle leg. Since the small sphere is tightly fitting, there is some time lag between a change in hydraulic resistance and the motion of the sphere. Experiment also with the effect of throttling the flow with your finger. Explain the reasons for what you observe. Change the flow rate and observe whether the bridge remains in balance or not. If so, why? If not, why not?
c) Re-establish a balanced bridge by returning the flow rate to the original setting. From this
point on, do not change the needle valve, as it is important for these next steps to be done at constant flow rate. Add an elbow to one of the legs and observe the resulting imbalance. Raise or lower one tube to re-establish the balance and record the elevation change necessary to accomplish this. This datum will be used to compute the minor loss using the mechanical
energy balance.
d) Using the empirical rule of thumb that an elbow creates a loss equivalent to 30 pipe diameters of smooth, straight pipe, shorten the leg containing the elbow by an appropriate length. Observe the bridge balance or imbalance. If imbalanced, measure the change in elevation of one of the tube exits required to re-establish balance.
Experiment Report
Pressure transducer
The Validyne pressure transducer produces a voltage related to the pressure difference across a thin plate. If the deflection of the plate follows the laws of linear elasticity, the pressure will be linearly related to the voltage and the device is said to be a linear
transducer. Perform a linear least‐squares analysis of the data. Try both linear and a
quadratic fits and compute the goodness of fit. Discuss the degree to which this is a linear transducer.
Paddlewheel flowmeter
The paddlewheel flowmeter produces a pulse signal, the frequency of which is related to the fluid velocity in the pipe. Perform a least‐squares fit of your data, using different polynomial fits and a power law relation. Find a suitable fitting function and record your fit. To what degree is the paddlewheel a linear transducer? Are there reasons to expect either linear or non‐linearity in the calibration? Discuss.
Orificeplate flowmeter The theoretical relation between Q and P is a nonlinear one, namely Q(const.)Cd P. Using log‐log scales, plot your data points for Q as a function of P. Do the data appear to fall along a straight line, indicating that a power‐law relation of the type m P K Q ( ) might apply? If so, what is m? If not, why not? Use the measurements of Q vs. P, compute the discharge coefficient Cd for all the data. Plot Cd vs. the Reynolds number Re and compare against standard curves. Head loss in pipe flow
Calculate the friction factors for each flow rate and tube size and plot all the data as a function of the Reynolds number. Use different plotting symbols for different tube diameters and check for Reynolds number scaling. Compare your data with the standard Moody diagram and discuss.
Minor losses
Express your results for minor losses through elbows, tees and couplings both as loss coefficients, K, and as equivalent lengths, Le/D . Compare your results with literature results for K, and with the common empirical rules of thumb for Le/D .
Piping network
Compute the loss coefficient and the equivlent length, Le/D , for an elbow as measured by the bridge technique. Compare it against your direct measurement and also against the standard rule of thumb.
